lesson thirty-one: what’s your angle?

LESSON THIRTY-ONE:

WHAT’S YOUR ANGLE?

WHAT’S YOUR ANGLE?

• Now that we have talked about inscribed figures, we can delve a bit more into angles within circles.

WHAT’S YOUR ANGLE?

• In a circle, there are infinitely many combinations of central angles.

• This is an angle whose vertex is the center of a circle.

WHAT’S YOUR ANGLE?

• The two arcs that are created when a circle is divided by a central angle are called the major arc and minor arc.

WHAT’S YOUR ANGLE?

• The minor arc is the one on the interior of the smaller central angle.

• This one has been labeled for you.

XC

AB

WHAT’S YOUR ANGLE?

• The major arc is the one on the exterior of the smaller central angle.

• Draw the arc on this circle below!

XC

AB

WHAT’S YOUR ANGLE?

• When naming the minor arc we need only two letters.

• The minor arc below could be named AB or BA.

XC

AB

WHAT’S YOUR ANGLE?

• The major arcs however, need three letters to be accurately labeled.

• ACB or BCA could be names for the arc below.

XC

AB

WHAT’S YOUR ANGLE?

• The angle of the major and minor arc will always be equal to the central angle which creates them.

WHAT’S YOUR ANGLE?

• When given one, you can find the other by simply, subtracting the measure from 360.

• Furthermore, you can find the sum of two non-overlapping arcs by simply adding their measures.

WHAT’S YOUR ANGLE?

• Sometimes, a circle be divided directly in half.• The result is two semicircles.• All of these have a measure of 180.• You may apply the same principles we just

discussed to semicircles.

WHAT’S YOUR ANGLE?

• For example, let’s see if we can find arc AC below.

X C

A

210

WHAT’S YOUR ANGLE?

• What about arc AB?

XC

A

B42

WHAT’S YOUR ANGLE?

• Aside from central angles, there are also inscribed angles.

• This is an angle whose vertex is on the circle.

WHAT’S YOUR ANGLE?

• How do you suppose central angles and inscribed angles are related?

WHAT’S YOUR ANGLE?• The measure of the inscribed angle will be

half of the included arc measure.• Furthermore, if two inscribed angle intercept

the same arc, then they are congruent.• Also, an inscribed angle that intercepts a

semicircle is a right angle.

WHAT’S YOUR ANGLE?• We will be able to use this information to

solve all kinds of problems.• See if you can find arcs AB and AC

40

CB

A

WHAT’S YOUR ANGLE?

• See if you can find arcs CA, BC and AB below.• HINT: You may have to draw on some old

knowledge.

60

45

B

C

A

WHAT’S YOUR ANGLE?

• Try this…find the central angle!

70

B

C

A

WHAT’S YOUR ANGLE?

• It will help you as you do these problems to fill in bits of the circle as you go!

• You might crack the code without even really knowing it!